Track1DTTHessianLichnerowiczResidualEntryFormulaReductionEndpoint
plain-language theorem explainer
Endpoint proposition for Track 1.D: raw scalar residual formulas equating Regge minus Lichnerowicz kernel entries to generator coefficients suffice to close the TT Hessian/Lichnerowicz consequence route. Discrete gravity workers packaging the spin-2 residual handoff cite it. Pure Prop abbreviation: formula data implies nonempty residual-row, kernel, match, and bilinear endpoints.
Claim. If residual-entry formula data for the period-5 TT Hessian versus lattice Lichnerowicz comparison is given (every residual kernel entry satisfies $\mathrm{Regge}(e,f)-\mathrm{Lichnerowicz}(e,f)=\mathrm{generatorCoeff}(e)(f)$), then there exist witnesses for residual-row coefficient entry data, residual-row coefficient data, residual-row span data, kernel residual TT-zero data, kernel row data, and operator match data, and the bilinear TT energy reduction endpoint holds.
background
Module is the Track 7 fork-handoff integration lane: it records what parallel endpoints prove without upgrading the discovery claim. Track 1.D sits in the tensor-shear sector, comparing the Regge second-variation TT Hessian on a period-5 edge lattice to the spin-2 lattice Lichnerowicz stencil.
Upstream residual structures package that comparison at increasing sharpness. Residual-row span data asks every residual row to lie in the combined conformal-plus-longitudinal generator span. Residual-row coefficient data supplies an explicit finite table of generator coefficients whose map recovers each residual row. The entrywise form strengthens this to scalar equality of every residual kernel entry with the corresponding generator-map reconstruction.
The bilinear reduction endpoint (already defined in this module) states that once Regge and Lichnerowicz operators match pointwise on TT modes, bilinear and quadratic TT energies match. Kernel-row data is the rowwise operator match on the longitudinal TT subspace. This definition packages the claim that raw scalar residual formulas alone unlock that whole chain.
proof idea
Definitional Prop, not a proved theorem. Antecedent is residual-entry formula data at $N=5$. Consequent is the conjunction of six nonempty structure witnesses (entry coeff, row coeff, row span, kernel residual TT-zero, kernel row, operator match) with the already-defined bilinear reduction endpoint. The companion ..._holds theorem discharges it by constructing each witness from formula data via the sector's ofFormulaData constructors, then invoking the bilinear endpoint.
why it matters
Closes the Track 1.D residual-entry handoff inside Track 7. Downstream, track1D_tt_hessian_lichnerowicz_residual_entry_formula_reduction_endpoint_holds asserts the Prop, and ForkHandoffIntegrationCert consumes the Track 1 reduction/interface package alongside Forks A--F (Schläfli stationarity, physical residual/Bianchi, many-body amplitude lift, Page-capacity transfer, $w(z)$ band, falsifier sensitivity).
Doc-comment states the scientific content: raw scalar formulas $\mathrm{Regge}(e,f)-\mathrm{Lichnerowicz}(e,f)=\mathrm{generatorCoeff}(e)(f)$ are enough to close the TT Hessian/Lichnerowicz consequence route. That is the sharp finite-stencil target for discrete spin-2 gravity in RS, feeding the structural master theorem's gravity lane without claiming full Schläfli leaf closure. Remaining open work is the displacement-class leaves flagged by the module doc.
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